The Study Of Largest And Smallest Order About Two Indexes On Some Kinds Of Three-cycle And Four-cycle Graphs

Commonly,the chemical and physical qualities of molecules can be statistically reflected by their topological index.Different topological index reflects different capability of the molecules.Especially,the Hosoya index and Merrifield-Simmons index are prominent topological indexes.Let G be a(molecular)graph.The Merrifield-Simmonsσ-index of G denoted byσ(G),is difined as the number of subsets of the vertex set V(G) in which no two vertics are adjacent in G,i.e,the number of independent sets of G,including the empty set;The Hosoya index is difined as the number of subsets of the edge set in which no two edges are adjacent in G,i.e,the matching humber of G,including the empty set.In the paper,we obtain the largest order of Merrifield-Simmons index on three-cycle which are attached by vertexes and the smallest order of the Hosoya index on four-cycle which are attached by vertexes.Besides,we also give the largest and the smallest order on the Hosoya index about three cycles of G which one cycle and two cycles having one common vertex are connected by one path.